oneoverr2 - a . or E < 0, k i is imaginary, and B/A is...

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1 /r 2 Potential Charles B. Thorn * (August 4, 2000) Put ¯ h = 2 m = 1. The radial Schr¨ odinger equation is - d 2 dr 2 - α 2 + 1 / 4 r 2 ! u ( r ) = Eu (1) The solutions are rJ ± ( kr ) with k 2 = E and J are bessel functions of order ± . Introduce a cutoF radius ± below which the potential is a constant continuous with the above at r = ± . Then the solution for r < ± is just sin( k 0 r ), with k 0 2 = k 2 + (4 α 2 + 1) / 4 ± 2 . Matching logarithmic derivatives of the wave functions at the cutoF gives k 0 ± cot( k 0 ± ) = 1 2 + AJ 0 - ( ) + BJ 0 ( ) AJ - ( ) + BJ ( ) (2) ±or energies such that k± << 1, the matching condition can be simpli²ed. Note that J ν ( z ) 1 Γ(1 + ν ) ± z 2 ² ν . as z 0. ±or ν = , z is periodic under z ze 2 π/α , so we can go to small cutoF by writing ± = ae - N 0 π/α with a ²xed and ²nite and N 0 → ∞ . The matching condition becomes independent of N 0 : q α 2 + 1 / 4 cot q α 2 + 1 / 4 - 1 / 2 = - A ( ka/ 2) - e ( α ) - B ( ka/ 2) e - ( α ) A ( ka/ 2) - e ( α ) + B ( ka/ 2) e - ( α ) (3) where we have put Γ(1 ± ) = | Γ(1 + ) | e ± ( α ) . ±or E > 0, is k real and this equation determines the phase shift in terms of
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Unformatted text preview: a . or E < 0, k i is imaginary, and B/A is xed by the condition that u = AJ i ( ir ) + BJ-i ( ir ) K i ( r ) is exponentially damped at large r , so the matching condition gives the energy quantization condition. q 2 + 1 / 4 cot q 2 + 1 / 4-1 / 2 = cot[ ln( a/ 2)- ( )] . We have immediately from the scaling periodicity that if is a solution so is n = e-n/ , where n is any positive or negative integer. In particular there are discrete energy levels extending to- . Of course with N nite the above solution breaks down for n -N and then one has to go back to the unsimplied matching condition to obtain the quantization condition. * E-mail address: thorn@phys.uf.edu 1...
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This note was uploaded on 09/25/2011 for the course PHY 6346 taught by Professor Staff during the Spring '08 term at University of Florida.

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